Senate Majority Leader Harry Reid of Nev., updates reporters on the pace of the immigration reform bill following a Democratic strategy session Tuesday on Capitol Hill in Washington.

WASHINGTON — Senate Republicans are split over the immigration bill steaming toward approval at week’s end, a divide that renders the ultimate fate of White House-backed legislation unpredictable in the House and complicates the party’s ability to broaden its appeal among Hispanic voters.

To some Republicans, the strength of Senate GOP support for the bill is all but irrelevant to its prospects in the House. Conservatives there hold a majority and generally oppose a core provision in the Senate measure, a pathway to citizenship for immigrants living in the United States illegally.

Any such impact is “greatly overrated,” said Missouri Sen. Roy Blunt, who previously served as chief vote counter for House Republicans.

But Rep. Paul Ryan, R-Wis., offered a different view. A Senate vote on Monday to toughen border security with thousands of new agents and billions of dollars in technology “obviously makes final legislation more likely,” the party’s 2012 vice presidential nominee said on CBS.

One prominent Democrat, Sen. Chuck Schumer of New York, also says House sentiment can be changed, particularly through the addition of strong border security measures of the kind that resulted from negotiations with previously uncommitted Republicans.

“I believe a large bipartisan vote will wake up our colleagues ... in the House,” Schumer said shortly before the Senate inserted a requirement for 20,000 new Border Patrol agents and a total of 700 miles of fencing along the border with Mexico.

“Hopefully, as congressmen look how their senators voted, they will be influenced by it.”

In the key Senate showdown so far, 15 Republicans voted to advance the legislation that toughens border security at the same time it creates a chance at citizenship for 11 million immigrants living in the United States illegally. Another 27 voted to keep the bill bottled up.

Republicans who voted to block the legislation generally did so after saying it would not deliver on its promise of operational control of the border.

“When you look at it, it doesn’t, and they know it,” Sen. Jeff Sessions, R-Ala., said of the bill’s backers, who quickly disputed the charge.

A political pattern emerged, as well.

Among Republicans who are seeking a new term next year and as a result face the risk of a primary challenge, only three voted with supporters of the measure. Eight did not, a group that includes the party’s two top leaders in the Senate, Mitch McConnell of Kentucky and John Cornyn of Texas, as well as Sessions, who has been one of the bill’s principal opponents across three weeks of debate.

While party leaders long have looked to immigration legislation as a way to broaden appeal among Hispanic voters, individual members of Congress report a different perspective.

“It’s hard to argue with the polling they’ve been getting from the national level,” Texas Republican Rep. Kenny Marchant said recently, referring to polls that show support for border security along with legalization. Yet in his own district in the suburbs west of Dallas, he said, proposals along the lines of the Senate bill are “very unpopular.”

The party’s potential presidential contenders also are split, likely a harbinger of a struggle in the campaign for the 2016 nomination.

Two of them, Sens. Rand Paul of Kentucky and Ted Cruz of Texas, oppose the legislation.

For his part, Cruz took a verbal poke at fellow Republicans in remarks on the Senate floor on Monday, saying that some senators in each parties “very much want a fig leaf” on border security to justify a vote for the measure.

Yet one Republican presidential hopeful, Sen. Marco Rubio of Florida, is a member of the so-called Gang of Eight, a bipartisan group that helped draft the bill. Among its provisions are several that impose conditions on immigrants seeking legal status, including payment of fines, pay outstanding taxes and undergo a background check.

In recent months, Rubio has sought to reorder the political circumstances rhetorically, asserting that the status quo amounts to “de facto amnesty” for those in the country illegally since it is unlikely they will be forced to leave. The phrasing marks an attempt to neutralize long-time claims that legalization confers amnesty. Increasing numbers of Republicans now employ similar rhetoric.

Among the unknowns is how much impact Rubio and the other Republicans in the Gang of Eight — Sens. Jeff Flake and John McCain of Arizona and Lindsey Graham of South Carolina — will have on House Republicans whose votes will determine the fate of legislation to overhaul the immigration system.

Rubio has met with members of the House Republican leadership as well as with Ryan and members of the conservative Republican Study Group.

Among House Republicans, supporters of legalization in any form, citizenship or otherwise, is scarce, although Blunt predicted there would be “an incredible amount of reasonableness” on that subject once lawmakers thought the border had truly been secured.

The House Judiciary Committee has approved two immigration bills recently, one of which echoes Mitt Romney’s suggestion in the 2012 presidential campaign that immigrants “self-deport” if they are in the country illegally. It encourages immigrants living in the United States to “depart voluntarily” at their own expense.

Neither of the bills cleared by the committee offers the prospect of legalization for immigrants in the country illegally, either citizenship or a step short of it.

House Speaker John Boehner, R-Ohio, has pledged not to bring legislation to the floor for a vote that does not have the support of at least half the GOP lawmakers in the chamber, a commitment made under pressure from restive conservatives that virtually rules out any measure envisioning legalization.

Some GOP lawmakers are hoping no immigration bill passes, to avoid the possibility of a final compromise with the Senate that goes further than they want.

Boehner also has said the entire House will “work its will” on the issue. It’s a comment that takes into account the potential impact of House Democrats, some of whom are already clamoring for a chance to vote on the bill that clears the Senate this week.

Republicans command a 234-201 majority, meaning that as few as 17 GOP defections could change the outcome of any vote.